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Brane Gravity and Brane Gravity and Cosmological ConstantCosmological Constant
Tetsuya ShiromizuTetsuya Shiromizu Tokyo Institute of TechnologyTokyo Institute of Technology
白水 White Water
Plan
1. Warped extra dimension - Randall-Sundrum braneworld -
2. Effective theory
3. D braneworld
4. Summary
Warped extra dimension
/ye
y
flat brane
/ye
constant alcosmologic negativebulk
Randall&Sundrum(1999)
Extra dimension
5-dimensions
The Einstein equation on brane Shiromizu, Maeda, Sasaki, 2000
gEGTRgR brane 482
1
4dim Einstein equation
Correction terms
22
525 6
1
2
1 bulkbrane
Cosmological constant on the brane
Bulk cosmological constant Brane tension
For flat brane, we assume Randall-Sundrum tuning 06
1 225 bulk
Two cosmological constants
Bulk cosmological constant
Cosmological constant on brane
Do not confused !Do not confused !
The Einstein equation on brane Shiromizu, Maeda, Sasaki, 2000
gEGTRgR brane 82
1
baba
brane
nnCE
TqTTqTTTT
G
)5(
2
22
222
24
1
8
1
12
1
4
16
18
6
1
2
1
Cosmology - FRW model -)( fluidPerfect uuqPuuT
))(2(12
1 uuqPuu
0)()(
0)( 0
PDuuguDuP
uDPDuTD
DuuqPD ))((6
1
0DIf homogeneous and Isotropic
0 ED
spatial derivative
fluid 4-velocity
Cosmology II
0 ,0
EDE
For the homogeneous and isotropic universe, it becomes the energy-momentum of “radiation”.
(Equation of state) (Equation of motion)
)(400 taE
(Dark radiation)
Equation is closed!Mass parameter of Schwarzshild-antideSitter Black Hole
Braneworld adS/CFT (Witten,Gubser,Garriga&Sasaki,Shiromizu&Ida,Tanaka,…)
Scale factor (~radius of universe )
Cosmology III
42
44
2
2
3633
8
aa
KG
a
a N
222
22
)()( Kdr
rf
drdTrfds
Friedmann equation on the brane:
Bulk spacetimes: 5-dim. Schwarzshild-anti deSitter spacetime
22
2
)(r
rKrf
(unit 3-dim. Sphere, plane or hyerboloid(K=1,0, or -1)
Langlois et al, Garriga & Sasaki, Ida, Kraus, Mukohyama et al, …
Problems
How to solve?
?E
✓Linear perturbation( Garriga & Tanaka, Sasaki, Shiromizu & Maeda,…)
✓Cosmological Perturbation( Kodama,Ishibashi&Seto, Mukohyama, Koyama&Soda, Langlois,…)
✓Numerical analysis (Shiromizu&Shibata, Kudoh,Tanaka&Nakamura,…)
✓Gradient expansion( Wiseman, Kanno & Soda, Shiromizu & Koyama,…)
✓Close limit of two branes (Shiromizu,Koyama,Takahashi, de Rham&Webseter,…)
✓Lower dimension (Emparan,Horowitz&Myers)
Model Randall-Sundrum I type
)()(2
2
1)(
)(4)(
)(4)5()5(52 branebrane LgxdLgxdRgxdS
)0( y
dxdxxygdyedxdxgds xybaab ),(2),(22
)( 0yy
y
- brane+ brane
Bulk action + Brane action -Brane action
Induced metric & extrinsic curvature
dcdb
caab nqqK
1 , baabbaabab nngnngq
ananan~
paralell transport
Induced metric
hypersurface
Extrinsic curvature
ay
a en )(
qeK
dxdxxyqdyeds
y
xy
2
1
),(2),(22
(1+4)-Decomposition
)(2
|)( )(2
TKK yy
04
3~
12~~
4
3
6 ,
4)(
~~
4
1
)(~~~
2)4(2
22222
traceless)4(
KDKD
RKKK
DDKKKKe
DDDDRKKKe
y
y
(Bulk)
(On Branes)
We solve the evolution equations along the extra dimension y
KKK
4
1~
Junction conditionsJunction conditions
2-1. Low energy effective theory
Wiseman 2002, Kanno & Soda 2002, Shiromizu & Koyama 2003, de Rham 2004
Small parameter
1~||
||2
2
)4(
brane
bulk
LK
R
),(),(),(
0),0( , ),()(),(),()1()0(
)1()1(2
xyKxyKxyK
xgxygxhxyaxyg
We are interested in long wave scale. Then small parameter is the square of ratio of the bulk curvature scale to the brane intrinsic curvature scale:
Sketch
RKy ~
branebrane RK |~|
The junction condition at the branebrane
brane TK 25~|
braneT 25~
Gradient expansion or close limit
braneTRhR 1252
1
0th order
04
3
12~~
4
3
4~~
4
1
0~~
)0()0(
2)0()0(2)0(
2)0()0(2)0()0(
)0()0()0(
KDKD
KKK
KKKKe
KKKe
y
y
2|)(
2)0()0(
yyKK
0th order
),(
)(),(
~
~~)0(
(0))0()0()0(
xyg
xCxyKKKKe y
0),(~
0)( 0x),(~
condition Junction (0)(0) xyKxCyK
ggeK y)0()0()0( 1
2
1
y
xyd
dyexydxhexyg0
),(2
)0( ),( ),(),(
“Integration constant”
0 ,12
4
3 )0(2
2)0( KDK
constraints
Junction conditions 2
6
11
1st order
TKK
KDKD
gRKK
eDeKKKe
eDDegRKKKe
yy
hh
y
y
)(2
)1(
)1()()1()(
)0()4()1()0(
2)1()1()0()1(
traceless)1()0()4()1()0()1(
2|)(
04
3~
)(2
3
)(2
1
)()(~~~
1st order
2)()()()()(2
2)()()(2
2)()()()4(2
)4(
)(2
111
)(212
)(1
)(
DddDdDdDDa
eDD
DddDdDdDdDDhRa
gR
hhhhhy
hhhhhh
traceless
)()()()(2)4(2)1(4 1)(
~)(
2
~dDdDdDDahRaKa hhhh
yyy
4
traceless
)()()()(2)4(2)1( )(1
)(~
2),(
~
axdDdDdDDahRaxyK hhhh
2)(2)(
2)4(
2)1( )(
11)(
6),( DddD
ahR
axyK hh
・ The traceless part of evolutional equation becomes
solution
Integration constant・ From the Hamiltonian constraint, we obtain
1st order
42)()()(2)()()(2)4(2
)1()1()1()1(
)(2
11)(
2
4
3~
aDddDdDdDdDDahGa
KKKK
hhhhhh
(2) )(2
)(2
(1) )(22
400
20
)4(20)(
)(2
)4()(2
adDDahGagT
hGT
40)2()1( a
)(2
112
)()()()1(
)()1(2
)()(2
0)(
0)(
0)(
02)(
0)()(2
0
)(20
)(2
)4(20
0)()(2
0)4(2
0)(2
0)(
2
DddDdDdDdDDa
hTahThGa
dDDahGahTahT
hhhhhh
hh
Effective gravitational equation
Junction conditions imply
Comment on RSII model
)(2
)( )(2
)4( xThG
undetermined
In principle, it is determined by boundary condition at “y=∞”
It may corresponds to the dark radiation, but we have to confirm that.
)()( 22
)4(
TOEThG
2-2. Close limit
Shiromizu, Koyama and Takahashi, 2003 (primitive version) de Rham and Webster 2005 (elegant version)
Assumption and motivation
Brane collisions is fundamental like particle collisions
Brane collision may give us a new picture of big-bang
1d
The brane distance is much smaller than the bulk curvature scale
What we have to do actually
)(2
12)( 2
2)4(
KKKKKKeeDDeyG y
can be written in terms of energy-momentum tensor on branes (Junction condition)
We must evaluate this
de Rham & Webster’s way
)0(!
1)1(
00
n
nyKn
yK
)/(1ˆ 2 dOKOK ny
ny
2,
,,
, )(:ˆ dSSddSddSO
)0(ˆ
)ˆsinh()0()ˆcosh(
)0()!12(
1)0(
)!2(
1
)0(!
1)1(
0 0
122
0
KO
OKO
Kn
Kn
Kn
K
y
n n
ny
ny
n
ny
)0()ˆtanh(
ˆ)1(
)ˆsinh(
ˆ)0(
K
O
OK
O
OKy
Effective equation
2)()()()()()(
)()()()()(
4)(
)(2
)(2
)()()(
72
1
12
1
8
1
3
1ˆcosech3
1ˆcothˆ
6
2
2
2
||tan
2
||tanh
1
2
||tanh||
11
TTTTT
TTOTTOOA
TTA
d
dddd
d
dd
ddDD
dG
Tensor perturbation
dd
dTh ijij
coth ,~ 22eff
2eff
2
Equation for tensor perturbation is same with 4-dim. one except for gravitational constant
3. D Braneworld
Shiromizu, Koyama, Onda & Torii, 2003 Shiromizu, Koyama & Torii, 2003 Onda, Shiromizu, Koyama & Hayakawa, 2004 Shiromizu, Himemoto, Takahashi, 2004 Iwashita, Shiromizu, Takahashi, Fujii,2005
D brane cosmology
Brane world is motivated by D braneD brane
deSitter/Inflation model in warped flux compactification
Kachru,Kallosh,Linde,Trivedi, 2003
Kachru, Kallosh, Linde, Maldacena, McAlister, Trivedi (KKLMMT), 2003
Gravity on D brane
D brane
55 SadS compactification
Tension T = Charge Q
Bulk: IIB supergravity compactified on S^5 (~ 5-dim. theory)
Z_2 symmetry
H_3=dB_2, F_3=dC_2, G_5=dD_4
B_2
D_4Brane: Born Infeld action + Chern Simons
From Ten to Five
334210
25
23
21
23
210
210
10210
10
4
1
|~
|4
1|
~|
2
1||
2
1||
2
1)(4
2
1
FHC
FFFHRegxdS
GFDCC
dedxdxxgds
HCFFHCFFdCFdBH pp
~~ ,~ ,~
)(
~ ,
~ , ,
540
25
22
325530331223
IIB Supergravity
Theory
5on ty supergravi IIB type:Bulk S
termSimons-Chernaction Infeld-Born :Brane
25
23
24
52
322)5(4
52)5(5
2|
~||
~|)(
2
1||
2
15)(
4
5)(4
2
1GFeHRegxdS
222244
brane 2
1)det( BBBCDQBhexdTS
form-5 form-3 S of size dilaton 5
324532323 ~~
,~~
,~ HCdDGHdCFdBH
Way to look at gravity on brane gK y~
) tensorstressbulk (~ RKy
branebranebrane RK |) tensorstressbulk (|~|
The junction condition at the brane )(~| 25
branebrane TThK
braneT 25~
Replaced by stress tensor on the brane using the junction condition for bulk form fields
braneT 25~
cancel
Gradient expansion or close limit
5-dim Einstein equation
Anti-de Sitter curvature radius
Gravity on brane
The gravitational theory at low energy
0~2
1RgR
Z_2 symmetry
H_3=dB_2, F_3=dC_2, G_5=dD_4
B_2
D_4B_2 on the brane is not the source of the brane gravity
Tension T = Charge Q
T ≠ Q
braneTTgQTRgR 112 )(2
1
rebraneTgQTRgR ,1212 )(2
1
branerebrane TTQTT 1, )(:
Tension T ≠ Charge QTension T ≠ Charge Q
Z_2 symmetry
H_3=dB_2, F_3=dC_2, G_5=dD_4
B_2
D_4
No cancellationNo cancellation
Cosmological constant appears at the Cosmological constant appears at the same time as B_2 is a source of the same time as B_2 is a source of the gravitygravity
Shiromizu, Koyama, Torii, 2003, Iwashita, Shiromizu, Takahashi, Fujii, 2005
Summary
Braneworld effective theoryBraneworld effective theory
Gradient expansionGradient expansion
Close limitClose limit
Braneworld based on D-braneBraneworld based on D-brane
The gauge fields on The gauge fields on BPS D braneBPS D brane is is not sourcenot source of gravity on brane. of gravity on brane.
The gauge fields on the The gauge fields on the deSitter (anti-) D branedeSitter (anti-) D brane is is sourcesource of gravity on brane. of gravity on brane.
Remaining Issues
Higher co-dimension, Warped flux compactifHigher co-dimension, Warped flux compactification( KKLT, KKLMMT model)ication( KKLT, KKLMMT model)
other fieldsother fields
stabilisationstabilisation
Basics of higher dimensional spacetimesBasics of higher dimensional spacetimes
Mukohyama et al, hep-th/0506050
Anti-D brane
Randall-Sundrum type modelRandall-Sundrum type model
T_+=Q_+ T_-=Q_-=-T_+=
D brane with positive tension
D brane with negative tension
Inflating anti-D brane in warped extra dimension
T_+=Q_+ T_-=Q_-≠-T_+
D brane with positive tension
D brane with negative tension
anti-D brane T_0=-Q_0
Koyama & Koyama, hep-th/0505256
cosmological constant is cosmological constant is induced on anti-D brane induced on anti-D brane
Note: D branes is not deSitter brane
If the brane distances are stabilised, D branes would be inflating
Gravity on anti-D brane?Koyama, Koyama, Shiromizu, Iwashita, to appear, 2005
Koyama & KoyamaKoyama & Koyama:
“anti-D brane is inflating”
Shiromizu, Koyama, Torii 2003/ Iwashita, Shiromizu, Takahashi, Fujii 2005Shiromizu, Koyama, Torii 2003/ Iwashita, Shiromizu, Takahashi, Fujii 2005:
“non-BPS brane is inflating at the same time as gauge field on the brane is a source of the gravity on the brane”
The gauge field on the anti-D brane The gauge field on the anti-D brane would be a source of the gravity on the would be a source of the gravity on the anti-D brane. anti-D brane.
Gravity on anti-D brane
)partraidon (22
1 25
0
25 braneTgTRgR
Koyama, Koyama, Shiromizu, Iwashita, to appear, 2005
Set-up
)(g
MNg
)(2B
)(2C
)(4D
23 dBH
23 dCF
45 dDG
)(2F
tenson=charge )( )(
)(2B
)(2C
)(4D
)(2F
tenson=charge )( )(
)(g
dxdxxygdydxdxgds NMMN ),(22
0~)(xy 0~)( yxy
brane-D brane-D
Toy Model
2||
2
1||
2
1||
2
1)(
2
1
2
1 25
23
23
2)5()5(52
GFHRgxdS
)(2
)(2
)(2
)(2
)(4)(
)(4)()brane(
)(2
)(2
)(2
)(2
)(4)(
)(4)()brane(
2
1)det(
2
1)det(
fffCDfqxdS
fffCDfhxdS
)(2/1)(
)()( ||
FBf
06
52 2
)(4 Background spacetime can be pure anti-deSitter+ flat branes
assumption
Equations
0~
2
1
~~
0~
2
1~~
0~
2
1
~
4
1~~
3
4
22
)5()5(2)4(
2)5()5(2)4(
yyyy
yyy
yy
yyy
yy
yyy
y
MMy
FHKD
GKG
GHFKF
GFKXX
KKTTRK
KTTRKe
MNKLAB
NMKLABMNKL
NMKLMNKL
NMKLMNNMMN
yyy
gGGFgFFHgHHgT
FHX
2 2 2)5(2 ~~
96
1|
~|
~~
4
1||
4
1)(
2
1
2
1
~
0~
0~
0
~~
4
3
2
1
)5(2
)5(22)4(
y
y
y
y
yy
GD
FD
XD
TKDKD
TKKKR
0~ FH
For simplicity, we assume
(Evolutional equations)
(Constraint equations)
Junction conditions
)()()(
2
)(2
)()(
2
)()(
2
)()(
2
)()(
8),(
),(~
2),(
~
),(
3
1
21),(
ffx
xG
fxF
fxH
TgxyK
y
y
y
y
branes-on D
)(
)(
)()()(
4
1 ffffT
Long wave approximation
12
L
scale curvature intrinsic brane :
scale curvaturebulk :
L
)1()0(
)1(2 )(),(),(
KKK
gxhxyaxyg
1st order
hhfeaxyF
feaxyH
hhGFa
H
hhGHa
F
y
y
yyyy
yyyy
)(62)2/1(
)(62)2/1(
)2/1(
4
)2/1(
)2/1(
4
)2/1(
2),(
),(
0~~
2
1
0~
2
1~
)(140
)()(60
)( TaTfaf
/0
0yea
)2/1()2/1(
,~
yy HF
brane-Don condition junction
brane-Don condition junction
1st order )1(K
)(~
)1(2
11
)(2
)(~
2
1
2
2)(
~
2
1
2
)(2
)(~
2),(
~
)(~~~
)4(20
)()()()(40
)()()()(
40
)(
160
2)4(2
0
)()()()()(
2
)(
2)4(
)()()()()(
2
4)(
162
2)1(
)(
1624)4(2)1()0()1(
hRaDDDDaDDDD
xaTahRaDDDDT
ThRDDDDT
xaTahRaxyK
TahRaKKK
tracelesstraceless
traceless
traceless
y
0)(1
)(1
)(3
2)(
3
2
)(6
)(1
)(3
2
)(6
)(1
)(3
2
)(6
),(
2220
22)()(
2)()(
2
)4(20
22)()(
2
)4(22)()(
2
)4(2
)1(
DDaDD
hRa
DD
hRDD
hRa
xyK
Traceless part
Trace part
Junction conditions
Junction conditions
Effective Equation at 1st order
2)(
)(
)()()(2
)(
)()(20
)()()(40)()(
2
)()4(2
0
2
112
)()()1(
DdgdDdDdDgdDDa
gagGa
0)(3
2)(
3
2)(
1 20)()(
2)()(
22)()(
220
adDdDa
Effective Equation at 2nd order
)()(2)(
)()()(
)()(20
140
1
)(20
120
140)()(
12
2)(
)(
)()()(2
)(
)()(20
)()()(20
40)()(
2
)()4(2
0
)1(12
1)1(
14
1
8
3
4
33
28
81
7
3)(
2
112
))(1(1)()()1(
TDTDDTDDaa
Taaa
DdgdDdDdDgdDDa
gaagGa
hhffa
aadDdDa
)()(2
0)()(
2
2)()(
20
20)()(
2)()(
22)()(
220
)1)((2
))(1(9
2)(
3
2)(
3
2)(
1